L(s) = 1 | + (1.83 − 3.18i)5-s + (2.63 − 0.277i)7-s + (0.301 + 0.522i)11-s − 5.25·13-s + (2.12 + 3.68i)17-s + (3.68 − 6.38i)19-s + (0.578 − 1.00i)23-s + (−4.25 − 7.37i)25-s + 7.97·29-s + (−1.57 − 2.72i)31-s + (3.95 − 8.88i)35-s + (0.00266 − 0.00462i)37-s + 4.01·41-s + 7.32·43-s + (−6.10 + 10.5i)47-s + ⋯ |
L(s) = 1 | + (0.822 − 1.42i)5-s + (0.994 − 0.104i)7-s + (0.0909 + 0.157i)11-s − 1.45·13-s + (0.515 + 0.892i)17-s + (0.845 − 1.46i)19-s + (0.120 − 0.209i)23-s + (−0.851 − 1.47i)25-s + 1.48·29-s + (−0.282 − 0.490i)31-s + (0.668 − 1.50i)35-s + (0.000438 − 0.000760i)37-s + 0.627·41-s + 1.11·43-s + (−0.891 + 1.54i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.167 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.167 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.262084253\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.262084253\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.63 + 0.277i)T \) |
good | 5 | \( 1 + (-1.83 + 3.18i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.301 - 0.522i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 5.25T + 13T^{2} \) |
| 17 | \( 1 + (-2.12 - 3.68i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.68 + 6.38i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.578 + 1.00i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 7.97T + 29T^{2} \) |
| 31 | \( 1 + (1.57 + 2.72i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.00266 + 0.00462i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 4.01T + 41T^{2} \) |
| 43 | \( 1 - 7.32T + 43T^{2} \) |
| 47 | \( 1 + (6.10 - 10.5i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.64 + 8.05i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.30 + 5.72i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.969 + 1.67i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.31 + 7.47i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.13T + 71T^{2} \) |
| 73 | \( 1 + (5.33 + 9.24i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.07 - 3.59i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 + (4.09 - 7.09i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.960399996312248142617336754670, −8.082386020593966108323853263136, −7.52921293435589712739029849677, −6.42191519714547151480536309266, −5.44181655522945305133366401506, −4.83496397576563383751182608840, −4.42748106753815925363385184379, −2.76950917195595313236018496480, −1.77562229888699374098735664837, −0.820198794240767689006107913509,
1.44631182567147438648308416865, 2.53493549341365392122818260526, 3.12596351646094277808361245902, 4.45432362332535465444527092482, 5.41524325196172355791759749559, 5.93976213192867028251138438182, 7.16919552866757851458624366044, 7.34725662090278102024782067752, 8.326999803679371742596315236295, 9.396699108102484905837243455809